Appl Math Optim 46:207–230 (2002)
2002 Springer-Verlag New York Inc.
Optimal Control of Diffusions
Hector O. Fattorini
Department of Mathematics, University of California,
Los Angeles, CA 90095-1555, USA
Abstract. We prove a version of Pontryagin’s maximum principle for time and
norm optimal control of linear diffusion processes. This result includes both neces-
sary and sufﬁcient conditions and implies a “concentration principle” for the optimal
Key Words. Linear distributed parameter systems, Optimal controls.
AMS Classiﬁcation. 93E20, 93E25.
Let ⊂ R
be a bounded domain with boundary , let A be a uniformly elliptic
operator Ay =
y + c(x)y (∂
and let β be a boundary condition on the boundary (either Dirichlet or of variational
type β y = ∂
y − γ(x)y = 0,∂
is the conormal derivative on ). For m = 1, 2, 3 the
∂y(t, x )
= Ay(t, x) + u(t, x)(t ≥ 0, x ∈ ),
y(0, x) = ζ(x)(x ∈ ), βy(t, x) = 0 (t ≥ 0, x ∈ )
models a diffusion in , .
If y(t, x) is a nonnegative solution of (1.1), then
y(t, x) is the concentration at x (at time t ) of the diffusing matter, so that
y(t, x) dx (1.2)
In the modeling of diffusions, c(x) = 0; however, the treatment is the same for c(x) = 0. See